Easy divisibility rules that hardly anybody knows (Part 1)

Most curriculum standards include divisibility rules for 2, 3, 4, 5, 6, 9, and 10. A few also mention rules for divisibility by 8 or 12. Divisibility rules for numbers like 7, 11, 13, 17, 21, and so on are usually omitted, but they should be included for four reasons:

The rules are useful.

They are easy to use.

It is easy to understand why they work.

Once students know why these rules work, they can easily discover new rules.

We will begin with an easy rule for divisibility by 11. In the next post, we will cover two more rules and explain why they work. The third post will cover 14 more rules.

The rule for divisibility by 11 is “chop, subtract.” You chop the ones digit and subtract it from what remains. If that difference is a multiple of 11 (0, 11, 22, 33, …), the original number is a multiple of 11. If that difference is not a multiple of 11, neither is the original number.

EXAMPLE 1: Is 682 divisible by 11? ANSWER 1: Chop the 2 and subtract: 68 – 2 = 66. Because 66 is a multiple of 11, so is 682.

EXAMPLE 2: Is 1014 divisible by 11? ANSWER 2: Chop the 4 and subtract: 101 – 4 = 97. Because 97 is not a multiple of 11, neither is 1014.

Sometimes you need to apply the rule multiple times, as shown in this example:

EXAMPLE 3: Is 32,741 divisible by 11? ANSWER 3: Chop the 1 and subtract: 3274 – 1 = 3273. But is 3273 divisible by 11? Apply the rule again: 327 – 3 = 324. Some people may see that 324 is not a multiple of 11, but to prove it, chop and subtract again: 32 – 4 = 28. Because 28 is not a multiple of 11, neither is 32,741.

PRACTICE: Prove that these numbers are all divisible by 11: 165 3,465 913 13,574. Then prove that these numbers are NOT divisible by 11: 183 705 3,422 26,435.